| L(s) = 1 | + (−0.207 + 0.978i)2-s + (0.665 − 1.59i)3-s + (−0.913 − 0.406i)4-s + (1.17 + 1.30i)5-s + (1.42 + 0.983i)6-s + (−1.84 − 1.89i)7-s + (0.587 − 0.809i)8-s + (−2.11 − 2.12i)9-s + (−1.52 + 0.878i)10-s + (3.24 + 0.688i)11-s + (−1.25 + 1.19i)12-s + (1.96 − 0.637i)13-s + (2.24 − 1.40i)14-s + (2.86 − 1.01i)15-s + (0.669 + 0.743i)16-s + (7.46 − 1.58i)17-s + ⋯ |
| L(s) = 1 | + (−0.147 + 0.691i)2-s + (0.384 − 0.923i)3-s + (−0.456 − 0.203i)4-s + (0.525 + 0.583i)5-s + (0.582 + 0.401i)6-s + (−0.696 − 0.717i)7-s + (0.207 − 0.286i)8-s + (−0.705 − 0.709i)9-s + (−0.481 + 0.277i)10-s + (0.978 + 0.207i)11-s + (−0.363 + 0.343i)12-s + (0.544 − 0.176i)13-s + (0.598 − 0.376i)14-s + (0.740 − 0.261i)15-s + (0.167 + 0.185i)16-s + (1.80 − 0.384i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.911 + 0.411i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.911 + 0.411i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.45756 - 0.313952i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.45756 - 0.313952i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.207 - 0.978i)T \) |
| 3 | \( 1 + (-0.665 + 1.59i)T \) |
| 7 | \( 1 + (1.84 + 1.89i)T \) |
| 11 | \( 1 + (-3.24 - 0.688i)T \) |
| good | 5 | \( 1 + (-1.17 - 1.30i)T + (-0.522 + 4.97i)T^{2} \) |
| 13 | \( 1 + (-1.96 + 0.637i)T + (10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (-7.46 + 1.58i)T + (15.5 - 6.91i)T^{2} \) |
| 19 | \( 1 + (1.28 + 2.88i)T + (-12.7 + 14.1i)T^{2} \) |
| 23 | \( 1 + (2.42 + 1.39i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (1.43 + 1.97i)T + (-8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (1.32 + 1.19i)T + (3.24 + 30.8i)T^{2} \) |
| 37 | \( 1 + (0.174 + 1.66i)T + (-36.1 + 7.69i)T^{2} \) |
| 41 | \( 1 + (-9.98 - 7.25i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 - 3.77T + 43T^{2} \) |
| 47 | \( 1 + (4.41 - 1.96i)T + (31.4 - 34.9i)T^{2} \) |
| 53 | \( 1 + (-5.09 - 4.58i)T + (5.54 + 52.7i)T^{2} \) |
| 59 | \( 1 + (11.9 + 5.31i)T + (39.4 + 43.8i)T^{2} \) |
| 61 | \( 1 + (6.11 - 5.50i)T + (6.37 - 60.6i)T^{2} \) |
| 67 | \( 1 + (-4.96 - 8.59i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (4.43 + 1.43i)T + (57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (2.80 - 6.30i)T + (-48.8 - 54.2i)T^{2} \) |
| 79 | \( 1 + (-6.86 - 1.45i)T + (72.1 + 32.1i)T^{2} \) |
| 83 | \( 1 + (-0.468 + 1.44i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (1.63 - 2.83i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (1.51 - 0.491i)T + (78.4 - 57.0i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.87381644269807851110424180905, −9.805351531449226513858163948059, −9.218273186487956740871418446764, −7.979119285757753994353827718399, −7.27015143225076769029342214783, −6.40676889927917106648735319242, −5.88543928574417619221125766556, −4.03786478647999851370301101739, −2.85945648126291324618052607133, −1.07453542935705337262745306711,
1.66865598126882329880423008120, 3.23010778100796206008402807373, 3.94911542103507108962811528350, 5.38879708946164615994549337155, 6.04126657594681627131775147122, 7.87674860862727854862486813079, 8.898778871569977263197268247661, 9.336914923087743156883728124620, 10.04488771710293237400127628392, 10.94755305741554802740927354494
